COBORDISM INVARIANTS, THE KERVAIRE INVARIANT AND … · 1. Universal manifolds, cobordism constructions and quadratic forms. In this section we introduce some general constructions

TRANSACTIONS OF THE

AMERICAN MATHEMATICAL SOCIETYVolume 178, April 197 3

COBORDISM INVARIANTS, THE KERVAIRE INVARIANT AND

FIXED POINT FREE INVOLUTIONS (!)

BY

WILLIAM BROWDER

ABSTRACT. Conditions are found which allow one to define an absolute

version of the Kervaire invariant in Z of a Wu-(<7 + 1) oriented 2(7-manifold. The

condition is given in terms of a new invariant called the spectral cobordism in-

variant. Calculations are then made for the Kervaire invariant of the zz-fold dis-

joint union of a manifold M with itself, which are then applied with M - P *, the real

projective space. These give examples where the Kervaire invariant is not de-

fined, and other examples where it has value 1 e "Z. . These results are then

applied to construct examples of smooth fixed point free involutions of homotopy

spheres of dimension 4/c + 1 with nonzero desuspension obstruction, of which

some Brieskorn spheres are examples (results obtained also by Berstein and

Giffen). The spectral cobordism invariant is also applied directly to these ex-

amples to give another proof of a result of Atiyah-Bott. The question of which

values can be realized as the sequence of Kervaire invariants of characteristic

submanifolds of a smooth homotopy real projective space is discussed with some

examples. Finally a condition is given which yields smooth embeddings of homo-

topy Pm's in Rm*K (which has been applied by E. Rees).

In [7], it was shown how to define an invariant for certain manifolds of even

dimension which generalized the Kervaire invariant for framed manifolds. In [8] it

was shown how to use this technique to define surgery obstructions. In this paper

we shall continue the study of this invariant.

First we consider the question of when the invariant is defined and give much

weaker conditions than those employed in [7]. For manifolds of dimension 2q

these conditions are shown to be B\v A cobordism invariants (using the notation

of [7]). We define a new cobordism invariant called the "spectral cobordism in-

variant" and show how to determine whether the Kervaire invariant is defined as

a function of this invariant. We then study some examples (such as sums of pro-

jective spaces), show when the Kervaire invariant is defined and when not, and

Received by the editors March 16, 1972.

AMS (MOS) subject classifications (1970). Primary 57D55, 57D90, 57D65; Secondary57D40, 57D15, 57E25.

Key words and phrases. Spectral cobordism invariant, Z -valued quadratic forms and

their radicals, Arf invariant, surgery obstructions, characteristic submanifolds of smooth

homotopy projective spaces, Brieskorn varieties, embeddings.

( ') Part of this paper was written while the author was visiting at the Faculte des

Sciences, Orsay, and at the IHES, Bures-sur-Yvette. The author was partially supported

by an NSF grant. This paper was originally submitted to Scripta Mathematica on September

24, I97O, and transferred to the Transactions after Scripta temporarily ceased operations.

Copyright © 1973, American Mathematical Society

193

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

194 WILLIAM BROWDER

use the results to show that (again in the notation of [7]) JL (v ,) = Z gener-

ated by P q. The same examples are used to construct smooth surgery problems, i.e.

maps into real projective space P q tot each q, such that the surgery invariants

are nonzero, for each q. We then construct examples of smooth fixed point free in-

volutions T on S + (a generator of bP + ) such that there are invariant

characteristic S C S + if and only it 4k + 1 - I = 0 (4) (an invariant sphere

E is called characteristic if it is the total inverse image of the standard S C

S + under an equivariant homotopy equivalence (S + , T) with ÍS k+ , A),

where A is the antipodal map). In particular this gives smooth examples in each

dimension 4k + 1 where the Arf invariant of an involution (see [lO]) is nonzero.

(This result has been obtained independently by I. Berstein [5] by a direct geo-

metric construction, and by Giffen [l3] using the Brieskorn spheres (see §6).)

Such examples were known previously in the piecewise linear case (see [l8] and

[30]) and in some special dimensions (see [2l]). We also study these examples

using the spectral cobordism invariant and obtain a lower bound on the number of

different examples one can construct this way. Finally we show that these ex-

amples are related to the involutions on certain Brieskorn spheres studied by

Atiyah and Bott [4] and get a new proof of their result.

In the final section we prove a theorem about smooth embedding of homotopy

projective spaces in euclidean space, which has been used by E. Rees [24].

Most of this paper was written before Brown announced his definition of a

Kervaire invariant in Z [ll]. His invariant is always defined, unlike ours, but

his is not as natural, involving a choice. All the explicit calculations done here

could be done with his definition, perhaps more simply.

1. Universal manifolds, cobordism constructions and quadratic forms. In

this section we introduce some general constructions for studying a cobordism

theory and use them to define a certain Z -valued quadratic form on certain sub-

modules of HîM; Z ) tot M a 2a-dimensional Poincaré duality space. This is a

review of results of [7] in many respects, but, in fact the constructions in [7] are

valid for Poincaré duality spaces, while here we shall give a more geometrical

definition valid for smooth or p.l. manifolds. In particular we give a somewhat

different way to define cobordism invariants in terms of a homotopy construction

different from the classical one of Thorn. We shall use it later also to study

fixed point free involutions. For simplicity we shall always talk about smooth

manifolds and linear bundles with the understanding that one can translate the re-

sults into the piecewise linear category using micro-bundles [l9l or block-bundles

([25L [22]) and p.l. immersion theory [l5]>

Let us consider closed manifolds M of a given dimension 222. Let B be a

classifying space for a theory of 22-plane bundles, n » m, y the universal


COBORDISM INVARIANTS AND THE KERVAIRE INVARIANT 195

72-plane bundle over B. A B-orientation of M will mean a linear bundle map of the

stable normal bundle v of Mm C Sm +n into y. If v and y were assumed to have

additional structure besides linearity, and the bundle map were required to pre-

serve it, then we would have a different notion of B-orientation (for example if y

and v were oriented in the usual sense) (see [7, §4]). A B-cobordism of two B-

oriented manifolds M. and M (with orientations b '. vQ —> y, fe,: v. —> y) will

be a cobordism W, dW = MQ—»M and a bundle map b : v —► y, where v is the

normal bundle of W C Sm +" x 7, with M. C 5m +" x i, and fe \v . = fe., i = 0, 1.z ' ' z z '

Now let ß, be a finite complex with the same homotopy type as B up to

dimension m + 2, for example take B , = (ttz + 3)-skeleton of B, if B if a complex.

Embed B, rectilinearly in S for some large Q, and take B2 to be a regular

neighborhood of B, in S . Let y~ be the inverse of the pull-back y2 of the

canonical bundle y to B 2 and let W be the total space of the disk bundle of

y2 over B ?. Then W is a manifold with boundary of dimension s. We shall call

W a "universal B-manifold." Such a construction was first defined by Milnor [191-

Now B2 is parallelizable as it is an open subset of S , so the tangent bun-

dle of W is p*iy2 + t )> where /): W —> B is the projection. It is easy to see

that the normal bundle of Ws in S +" is the pull-back of the canonical bundle y,

to W, i.e. p*(y2).

If Mm is a B-oriented manifold, fe: v —> y, then since W is the same homo-

topy type as B up to dimension ztz + 2, we get a unique map fe2: v —» ¿>*(y2) where

y2 is the pull-back of y to B 2- Then fe.,: z^ —> p*(y2) defines a unique map fe~ :

i/" —> p*iyA~ , and v~ - r + f the stable tangent bundle of M. Hence we get

a bundle map c: r + í + ey —> p*(y2)~~ + f , c = b~ + 1, i.e., a linear bundle

map of the tangent bundle of Mm x D in the tangent bundle of Ws, s = m + k,

k = I + Q. By Hirsch's Theorem [16] such a bundle map comes from the differen-

tial of an immersion a: M x D —> W, which is unique up to regular isotopy. As k

is very large compared to m, we can change a by a regular isotopy to be an em-

bedding on M, and hence a small neighborhood of M, i.e., on a smaller M x D .

By a similar argument this embedding will be unique up to an isotopy. Thus we

have shown

(1.1) Theorem. Leí Wm + fee the universal manifold constructed above, and

Mm a closed B-manifold. Then the B-orientation of M defines an embedding

M x D C Wm + which is unique up to isotopy. Further, a B-cobordism U between

M and M leads to an embedding of Um +1 x Dk C Wm +* x /, with restrictions to

M.x D the given embeddings, for i - 0, 1.

Given a B-manifold M, and the embedding Mm x Dk C Wm+k, we can take the

Pontrjagin-Thom construction, i.e., consider the natural collapsing map g: W/dW

—♦ M x Dk/M xS'"' = 2feM+, the 4-fold reduced suspension of (zM U point). We


196 WILLIAM BROWDER

call g the "spectral orientation map" of M.

(1.2) Corollary. // Mm is a B-manifold, W a universal manifold as above, then

the spectral orientation map g: W/dW —> 2 M+ z's uniquely defined up to homotopy.

For an isotopy of the embedding leads to a homotopy of g.

From the fact that X M + is ik — l)-connected and W/dW is of dimension

m + k, it follows that, since k is larger than 222 + 1, all the homotopy information

carried by g will be the same, independent of the choice of S. in our construction

of W.

The spectral orientation map g will play an important role in defining the

quadratic form which yields the generalized Kervaire invariant. We recall that in

[7] a more general construction was given for M a Poincaré complex, etc., using

Spanier-Whitehead duality instead of immersion theory and the Pontrjagin-Thom

construction. In the sequel, therefore, we could use this more general situation

to enunciate the theorems, thinking of the construction given here as an intuitive

special case.

Taking the projection map M x D —> D and the induced map t: I Mt —»

S , and composing with g,

W/dW -I» lkM+-L Sk,

we get an element in the cohomotopy group 27 (W/dW). Using the usual transverse

regularity argument we then get

(1.3) Corollary. The B-cobordism group of m-manifolds is isomorphic to

it (W/dW), where W is the universal B-manifold constructed above, dimension W

= m + k, k > m + I.

This corollary is just the Spanier-Whitehead dual of Thorn's theorem that the

cobordism group is isomorphic to 27 ._(T(y )), where y is the canonical q-

plane bundle in the B-theory.

Instead of taking simply the projection map M x D —> D , let us instead

take ex 1: M x D —► W x D where c: M —» W is the classifying map for the

normal bundle of M, considering W as the classifying space in dimensions < 222

+ 2. The induced map 2kic): 2^/Vl + —» 2% + may be composed with g: W/dW—>

lkM+ to get h = (ïk(c))g: W/dW -> lkW +.

(1.4) Corollary. The homotopy class of h is a cobordism invariant, i.e., if

M. is B-cobordant to M. then hQ is homotopic to h,.

(We shall call the homotopy class of h £ lW/dW, 2kW +], the spectral co-

bordism invariant of the manifold M. )



Proof. From (1.1), we get a map G: W/dW x / -» 2kU + with G\W/dW x i = g.

the spectral orientation map of M., g.: W/dW —» 2 M. , z = 0, 1. Then the clas-

sifying maps c.: M. —> W extend to C: U —» W with C/M. = c, z = 0, 1. Thus the

composite S (C) o G: W/dW x / —* S W+ and clearly induces a homotopy of gn

with g,.

As lkW+ has Sfe as a retract, [W/dW, 2*W+] has nkiW/dW) as a retract, and

in view of (1.3), [W/t?"/, £ W^ is in general much larger than the cobordism group.

However, the richer structure of W in cohomology, as compared to S , makes it

convenient for defining certain cobordism invariants which we shall do in §2.

Now let iU, dU) be a Poincare duality pair mod 2 of dimension tz, i.e., there is a class

[U] £ H iU, dU; Z2) such that [U] n HKU; Z.,) — Hn_ .((/, dU; Z2) is an isomorphism for

all z. This is easily seen to be equivalent to the statement that the pairing H'iU; Zj®

H"~'iU, U; Z2) —> Z given by (x, y) = (x u y)[U], is nonsingular for all i. Then

the Wu class v.iU) £ H{iU; Z2) is defined by the formula iv.W), x) = (Sq'x)[(7],

for all x £ Hn~'iU, dU; Z2). It was proved by Wu that v ¡U) is a certain poly-

nomial in the Steifel-Whitney classes in case M is a manifold, so that a universal

Wu class v. £ HliB ; Z A (or more generally v. £ HliB G; Z A where BG is

classifying space for spherical fibrations). We shall take B to be the classifying

space for a bundle theory in which iz , = 0. (A universal construction for such

a theory is given in [7, §2].) Then the construction above, starting from such a B,

gives a universal manifold W such that (Sq? + \x))[W] = 0 for x £Hs-"~\w, dW; Z.,).

(See [7] for a more complete discussion.)

Suppose M q is a manifold oriented in a theory where v , = 0 and let

g: W/dW —►£ M+ be its spectral orientation map. Define 77 to be the composition

k *

HHM; Z2)^~>Hq+kilkM+; Z2)-^Hq+kiW/dW; Z2).

Following [7] we define a map ifi: (ker 77 )q —> Z as follows:

Let a: M +^> KiZ2, a) be such that a*it) = x, 1 £ HqiKiZ2, q); Z2) the funda-

mental class, KiZ q) the Eilenberg-Mac Lane space with homotopy groups

77. = 0 if i /= q, tt = Z Define ß to be the composition

V//dW-J^lkM + ̂ -^lkKiZ2, a).

If rjXx) = 0, then ß*ilkit)) = g*akaiù) = g*ïkix) = r,*x = 0. Hence we may de-

fine the functional Steenrod square

H2q+kiW/dW; ZAs^yHiHt)) £_L__L_i .

(imSq9+I + im ß*)


198 WILLIAM BROWDER

It is shown in [7, §l] that the denominator (im ß*) 2q+k = 0, im Sq9+1 = 0 by

assumption, so that Sqqß+l(2k(t)) £ H2q +*( W/dW; Z>>. Define ifrix) =

(Sq^+ (1 it)))lW], Note that on the right-hand side, only the map ß depends on x.

In [7, §l], it was shown that if x, y £ (ker 77 )q then ifrix + y) = i/)(x) + t/z(y) +

(x, y), i.e., that xp is a quadratic form over Z

2. Definability and additivity of the generalized Kervaire invariant. Let M2q

be a 2<7-dimensional (mod 2) Poincaré duality space and

W/SW-^~>2kM+^-£ 2kW +

be maps, with g the spectral orientation map defined by the W-orientation of M and

c the classifying map for the Spivak normal fibre space of M (or normal bundle if

M is a manifold), W being a "universal manifold" for the fibre space (or bundle)

theory with v , = 0 (see §1, and [7]). Recall that the homotopy class of h =

(2 c)g depends only on the cobordism class of M, and is called the spectral co-

bordism invariant of M (see §1).

Then as in §1 (see also [7]) we have a quadratic form defined by ip: (ker 21 )q

-+Z2 (where as in §1, 27*= g*2k: Hq(M; Z2) — Hq^(W/dW; Z.,)), and the asso-

ciated bilinear form

(x, y) = i/>(x + y) - i//(x) - i/)(y)

is the intersection form (x, y) = (x U y)([/Vf]), where [M] e H2 ÍM; Z 2) is the funda-

mental class. In order to define the Arf invariant of a quadratic form it is neces-

sary that the associated bilinear form be symplectic, that is, that it be non-

singular and (x, x) = 0 for all x. For the bilinear form associated with a quadratic

form over Z the second condition is automatically satisfied, i.e., (x, x) =

ipi2x) - 2yj(x) = 0. In particular it follows that (x, y) = (y, x). But the condition

of nonsingularity is not automatic and will not be satisfied in some examples we

shall develop later.

If ( , ) is a symmetric bilinear form on a vector space V, we define the rad-

ical R of ( , ) by R = ir e V such that (r, x) = 0 all x e V\. If t/f is a quadratic

form ifr: V —> Z with ( , ) its associated bilinear form, V a Z2 vector space,

we shall call i/z proper if ipir) = 0 for all r £ R = the radical of ( , ). If ip iS

proper, then i/z defines a quadratic form ip': V/R —* Z2 by yj'ix) = i/>(x) for x e V

representing x e V/R. Then x + r, r e R also represents x e V/R, and

i/z(x + r) = xffix) + ipir) i- ix, r). Then (x, r) = 0 since r £ R, if/(r) = 0 since i/z is

proper, and thus i/V: V/R —'Z is well defined. But the induced bilinear form on

V/R is nonsingular so that the Arf invariant of yj is defined. In this case one

may define the Arf invariant of i/z as the Arf invariant of yj , and we have proved

(compare [7, (III. 1.14)]):



(2.1) Lemma. The Arf invariant of \¡) may be defined if yj is proper, i.e., if

vj annihilates the radical of its associated bilinear form.

Now we have the following theorem about the geometric situation.

(2.2) Theorem, c H (W) is the annihilator of (ker q ). /tí particular, if R is

the radical of the intersection form on (ker r¡ )q, R = (ker q )q Pi c HqiW; Z A.

First we prove the following lemma.

(2.3) Lemma. Let U be a compact n-manifold with boundary, i: U —> W be an

embedding in the interior of W, where W is a compact n-manifold with boundary.

Let g: W/dW — U/dU be the natural collapsing map. If z £ HliU, dU), x e

H"-'iW), then

iz u i*x)[U] = ig*z u x)[W],

where H*iW/dW) and H*iU/dU) are identified with H*iW, dW) and H*iU, dU)

respectively.

Proof of (2.3). Consider the following commutative diagram:

(2.4)

(ty, dU)-

à (A)

- (W, W-int U) «-1 iw, dW)

iU, dU) x U — iW, W-int U) x WJAA (W, dW) x W

(B)

yxl

where e is the map of pairs induced by i, j is the natural inclusion, and A is

the natural diagonal map in each case, which induce the cup products in co-

homology. Now e is an excision, so e^ and e are isomorphisms, and the diagram

■* - 1

H*iW, dW) -L-+ HAW, W-int u) li— hAu, du)

HîW/dW)a*

Hîu/dU)

is commutative, together with the similar diagram in cohomology.

Now considering square (A) in the diagram (2.4), we find that e ie (z)u*)

= e (e iz)) \jix-z\jix. Considering square (B), we get

j*ie*-Hz)ux)= Í*ie*-liz)) ux.

Hence / e \z U t x) = g (z U / x) = \g z) U x.


200 WILLIAM BROWDER

Now lU] = g [W] so that

(z U i*x)lU] = (z u !-*x)(gJW]) = ig*iz u i*x))lW] = ig*z Ux)lW],

which proves (2.3).

Applying (2.3) to the situation of Theorem (2.2), we let U = M x Dk, W = W

so that i = c, and g = g. Then it x £ H iW) and y e /7*(M), we have

iy, c*x) = iy u c*x) [M] = (2*=y u c*x) [M x D¿]

(2"5) =(g*2fey ux)[W] = (77*y ux)[W].

Hence it follows that (ker r¡ ) annihilates (im c ), so (im c ) C u = annihilator

of (ker 77 ).

Since H (M) ^ ker r] + im 77 , it follows that rank (im c ) < tank (im rf) =

rank Ü. Similarly, (im 27 ) C annihilator of (ker c ) so rank (im 27 ) <

rank (im c ) = rank of the annihilator of ker c . Hence rank (im 27 ) =

rank (im c ) and it follows that c H (W) = U, the annihilator of (ker 27 ). Then

R = Q n (ker 77*)* = c*HqiW) n (ker rj*)q, and (2.2) is proved.

(2.6) Corollary. The Arf invariant of ip is well defined on (ker 27 )q if and

only if yb I c*HqiW; Z.,) n (ker 27*)« = 0.

We recall that i/z is defined in § 1 using the functional Sq9 + associated to

the composite

W/dW-L* I4Mti-il*K(Z2,,)

where x = a (z) e (ker 27 )q. If x = c iz), z £ HqiW; Z2), then a factors through

8: W —» KÍZ q), so that a= Sc, and we have

W/dW _£-, 2feM+—. 2%+ — 2feK(Z2a).

Now (2 c)g = A, the map defined in §1, whose homotopy class is the spectral

cobordism invariant of M.

Thus, if x e (ker rf*)q n c*HqiW; Z2), x = c*y, the map ß: W/dW —

1 K(Z (7) corresponding to x is the composition ß = (2 5)6, where ¿: W/dW—>

2 W + is the spectral cobordism invariant of M, 8 it) = y, and ¿ y = 0. Thus

Sq|+1(2fe(t)) = Sq9*1 (2fe(z)) and it follows that xf/ix) = 0 for all x £

(ker 27*)« n c*HqiW; Z,) if and only if (Sq,+t\w (2*(t)))[W] = 0 for all S: W + —

KiZ2, q), such that 5*z £ ker £*C f/«^(2feW; Z2). This condition depends only

on h, so we get the following theorem.



(2.7) Theorem. Ler M q be a Poincaré duality complex of dimension 2a,

oriented in a theory which has v , = 0. Then for the quadratic form \p: (ker r\ )q

—»Z , defined by the orientation, the Arf invariant of i/z is defined if and only if

(Sq«*,1 ilkt))[W] = 0 for all 8: W. — KiZ q) such that 8*i £ ker h*lh n

HqiW +; ZA, where h: W/dW —> S W+ z's the spectral cobordism invariant of M isee

§1). In particular, the property that the Arf invariant is defined is a cobordism in-

variant (in a theory with v , = O).

If the hypothesis of (2.7) is satisfied then we set kiM) = Arf invariant of if/,

and we call it the Kervaire invariant of M. By [7, (3.2)], this is exactly Kervaire's

invariant in the case of framed manifolds.

(2.8) Corollary. With M2q as in (2.7), suppose also that h*iH2q+kHkW +; Z2))

= 0. Then kiM) is defined if and only if Sq?+ (S y)[W] = 0 for every y £

ker h*lk nHqiW+; Z 2).

Proof. Note that if S i = y, then

Sqq+l ilkL) = Sqq + 1ilky) modh*iH2q+kiIkW + ;ZA) = 0.(2fe3)/, * 2

Hence the equality holds and (2.8) follows from (2.7).

Now let us consider the operation of addition in our cobordism theory, which

is represented by taking the disjoint union of the manifolds, so that M. U M. rep-

resents the sum of M , and M-. Suppose we have i.: M.x D CW and z' : M. x

Dk C W. Then we may consider z', x 1/3: M, x Dk +1 C W x I and z'2 x 2/3: M2 x

D + C W x I, so that the embeddings are disjoint in W x I and W x I is still a

universal manifold. Then the spectral orientation g: W x 1/diW x /) —»

2k+1iiMJ U M2)+) can be factored through W x I/diW x /) — W x //(rXWx /) U W x >/0,

and W x I/idiW x I) U W x V2) = ÍW x I/diW x /)) V (W x //<9(W x /)). In fact W x

I/diW x I) = 2(W/<9W) and this is the natural map 2iW/dW) — HW/dW) V S(W/f9W).

Also (M. U M ), = M, V M, , so 2fe+1(M. u/M,)+=Sfe+1M. V It + 1L and1 2+ I + 2 + ' 1 2+ 1+ 2 +

g: KW'/áW) -Sfe+IM,+ V I^+1M2+ is g = g,' + gz\ where g/ is the composite

2("7/r3W) —i, 2fe+1M.+ — 2^+1M,+ V M2+, g. being the spectral orientation for

M ., and + is the addition of maps induced by the suspension structure of ¿LiW/dW).

Now we return to the previous notation with g: W/dW —> 1, (M , U zVO+, with the

understanding that everything is a suspension, g = g. + g2 etc. as above, and we

have shown:

(2.9) Lemma. Let g.: W/dW —» rM. be the spectral orientation maps for

M ., i = 1, 2. Tfeerz the spectral orientation map for M, u M z's given by g - g' +

z?-! (/?•' = z'z?'. ^feere z. is the inclusion i :. 1kM. — i1kM, ) V (2feAl ), /'= 1, 2


202 WILLIAM BROWDER

and 2 c = 2 c, V 2 c2). Hence h = h, + h2 and the spectral cobordism invariant

is additive on cobordism classes.

Since ker ih 1 + h 2 ) = ker ih, + h 2) is larger in general than ker h, + ker h* ,

the condition that the Kervaire invariant is defined for two out of three for M =

AI, + AL is not enough to ensure it is defined for the third. We will see some ex-

amples of this in §3. In fact we will give examples of M, and AL, where (a) it

is defined for Al, and M2 and not for M + AL, (b) it is not defined for M. or M

and defined for M, + M2, (c) it is defined for Al,, not for AL, and either defined

or not for Al, + zYL. Thus all possible situations occur.

Hence in order to state an addition theorem for the Kervaire invariant we must

make assumptions that it is defined for each space. We give an additivity theorem

below (2.13), but it is unclear what are the most general hypotheses under which

additivity holds.

Let M. and AL be Poincaré duality spaces (mod 2), oriented in a theory

where v ,=0, A1 = A1,+AL (notation as above).q+1 ' 12 '

By (2.9), g = g[ + g¡: W/dW — 2kM x + V ^kM2 + , where g>. is the spectral

orientation g.: W/dW —» 1kM. followed by the inclusion in 2*AL V 2^/VL . It°7 I + ' 1 + 2 +

follows that (ker g*) 3 (ker g* ) + (ker g * ), and in general the inclusion may be

strict (which causes all the difficulties, in the definability of the Kervaire invari-

ant being nonadditive, and in proving additivity).

Let c: (M. U ^0+ —* ̂ + be the classifying map for AI, U M so that

c | AL = c, the classifying map for AL, 2*c: 2*M,+ V 2feM2+ —• 1kW + is the sum

1kc = 2fec, + 2fer Then im c* = im (c* + c* ), but note that of course this is

not the sum of the two images, in general.

Set 77*= g*2fe, as usual, so that 27* = 77* + 27*, where 27* = g*2 , and as

above (ker 27*) 2 (ker 17* ) + (ker 27* ).

Let yj be the quadratic form for M , + M2, if/. the quadratic form for AL, i =

1, 2 (as in §1).

(2.10) Lemma. O22 (ker 27*)* + (ker 27*)", xfj = i/z, + xfi r

Proof. Since H*iMx; Z_) and H*(M2; Z2) ate orthogonal in H*iM; Zj), it

follows from the fact that ifi is quadratic that ipix, + x2) = (//(xj) + i/z(x 2) if x. £

ker 77*, i = 1, 2. If /: M —» K(Z2, a) is such that /*(t) = x. £ ker 27*, i = 1 or 2,

rf. /.then / factors: M-^ Al. -1* K(Z2, a) where ¿¿: AI —» AL is the identity on AL and

the constant map on the other component, /*(() = x. £ HqiM.; Z2). Then (2 t/.)g =

(2¿a'.)(gJ + gp = g¿, the spectral orientation for AT (using (2.9)) and hence

Alk'f) = g.f. and Sq«+1t ilki) = Sqq\lilki) and hence ibix ) = xfj.ix ) and the5 ' 6i'i g(Zk/) Si'i ' ' '

lemma follows.

We get as a simple consequence



(2.11) Proposition. // (ker 77 )q = (ker r¡ Aq + (ker rj )q, and if the Kervaire

invariant is defined for M and M., then it is defined for M + M and

kiMl + m2) = MMj) + Mm2).

Proof. The radical is the sum of the radicals, so yj = if/. + ip2 annihilates the

radicals, and kiM + M2) is defined. Hence, from (2.10), kiM + M A = kiM A +

kiM2).

(2.12) Corollary. // T¡*HqÍM.; Z.,) = 0, i= 1, 2, then k is defined for M ,,

zVf., arza" M + M and is additive, i.e., kiM. + M A = kiM A + kiM A.

We would like to have more delicate results than (2.11), but it seems dif-

ficult to obtain a general theorem. However, we may prove

(2.13) Theorem. Suppose the Kervaire invariant is defined for M. and M.,

and that fe. = 0 on HqiW; Z A. Then the Kervaire invariant is defined for M +

M2 and kiM, + M2) = MM,) + kiM2).

(2.14) Corollary. Suppose kiM A, kiM A, and kiM , + M A all defined and sup-

pose fe* = fe* o/z HqiW; Z2). Then kiM, + M.,) = kiM,) + kiM2).

Proof of Corollary (2.14). Under the hypothesis, h = fe, + fe2 = 2fe, = 0 on

HqiW; Z 2). Hence by (2.13), we have ki- M, + (M, + M.,)) = ki- M ,) + kiM, + M 2).

But ki- M A = kiM A, obviously, and - M , + (M + M A = M Hence we have

kiM A = kiM A + kiM. + M A, and shifting kiM A to the other side of the equation

gives the result.

Note that in general - M may not be cobordant to M in a cobordism theory

with v , = 0, as we shall see in §3 for P q, the real projective space. But the

spectral orientation and classifying maps differ by automorphisms, so that the de-

finability and value of the Kervaire invariant is the same for M and — M.

(2.15) Corollary. Let M2q be such that (image c*)q C (ker 77*)" and kiM) is

defined. Then kinM) is defined for all n and kinM) = nkiM).

Proof, (image c )q C (ker q )q implies that fe = r) c =0, and hence fe =

nh = 0, where fe is the spectral cobordism invariant for tzzM. Since kiM) is de-' 72 r

fined, Theorem (2.13) applies to prove (2.15) by a simple induction.

Proof of Theorem (2.13). Since fe* = 0 on HqiW- Z.,) and kiM?) is defined,

it follows that Sqq + lilki)[W] = 0 for r= (I*a)¿2, any a: W+ — KiZy a). Since

MM,) is defined, Sqf + KSMtvV] = 0 for t = ilkß)h,, ß: W+ — KiZ 2> a) such that

ß*it) £ (ker fe*)9.

Now fe = 0, and fe = fe, + h implies that (ker fe )q = (ker h Aq. Now

(X^Xfe, + fe2) = ilkß)hl + ilkß)h2, so that if x £ (ker fe*)«, a= /3: W + —

fC(Z2, a) such that a*(z) = ß*{i) = x, then !>(*) = (Sq« + 1(2t(t)))[H'] =

(Sq'+KÂ) + Sq«+1(2*(t))[W] = 0 using [7, (1.6))]. Hence ¿(M, + M.,) is defined, by (2.7).


204 WILLIAM BROWDER

Now ker 27 D ker 27, + ker 272, and if we set R . = ker 77. n image c. = radical

of ( , ) on ker r¡., by (2.2), then ker r]{ = R¿ + Bp where ( , ) is nonsingular on

B ., z = 1, 2. Then we may choose a complementary subspace N to ker 27. + ker 27.

in ker 27 such that N is orthogonal to B . + B 2 (by subtracting off elements of

B , + B 2, if necessary). Then the Arf invariant ciyj) = ciyj ) + ciyj.) + ciyj') where

ii7 is i/z I (R , + R + N). Thus to prove the theorem we must show ciyj') = 0.

Since b = r¡ c = 0, it follows that im c Cker r;,, so that fi = ira c,.

Now 27* is injective on N. For if (x, y) £ N C tfÂI,; Z 2) + H9iM ; Z ), and

272(x, y) = 0, i.e., y e ker 772, then 27 ix, y) = 77, x + 272 y = 27 , x = 0 so x e ker 27*

and (x, y) £ ker 77, + ker 772. But N O (ker 77 x + ker 272) = 0, so (x, y) = 0 and 27*

is injective on N. Then it follows from (2.5) that z —> (2, ): N —» Z defines a

map of R2 = im c2 onto Horn ÍN, Z ) so that N n R = 0, where R = radical of

( , ) on ker 27 . Since y/' | R, + R2 = 0 and R, + R2 is a self-annihilating sub-

space under ( , ), it follows that we may choose representatives a., b. £ ker 27

of a simplectic basis of ker 27 \R so that a. e im c2 = R2, b. £ N. Since yj' \ R?

= 0, it follows that ciyj') = 0, which completes the proof of (2.13).

We note in passing the following property of h.

(2.16) Proposition. Let x £ H iW; F), F a field. Then h x = 0 if and only if

all characteristic numbers involving x are zero.

Proof. Suppose y £ H*iW; F) such that ic*x, c*y) = iic*x) U (c*y))[/Vl] / 0.

By (2.5), (c x, c y) = (27 c x, y) = (27 c x u y)lW] ¡= 0. Hence h x = 27 c x / 0.

Suppose ¿ x / 0. By Poincaré duality in W, there is a y £ H (W; F) such

that (A x, y) j¿ 0. But A x = 77 c x, and (27 c x, y) = (c x, c y) by (2.5) so

(c*x uc*y)lM]/ 0.

Thus the induced cohomology map of the spectral cobordism invariant h may

be interpreted in terms of characteristic numbers. But more subtle properties of

h, such as the functional Sq9 + of (2 o)b, a: W + —» K(Z2, q), may be harder to

interpret in terms of classical invariants.

3. Multiples of a manifold. In this section we study multiples of a fixed mani-

fold Al. By (2.14) if MAI) and 7e(2Al) are defined then ze(2Al) = 2kiM) = 0, and by

(2.14), M22M) = ?7Ze(M), so we wish to study the question: When is ze(2Al) defined?

We will give a condition (which is independent of whether /«(Al) is defined or not):

ki2M) is defined if and only if, in Al, v (Al) £ Vet 17 , or equivalently, if and only

if all characteristic numbers of Al involving v (Al) are zero (Corollary (3-9)). Next

we consider how to calculate ze(2Al) when MAI) is not defined and we show that

if MAI) is not defined but M2Al) is defined, then M2A1) = ipiv ÍM)) (calculated in

Al), (Theorem (3.10)). This yields the cotollary (3.11) that M4A1) is defined and

ki4M) = iv (A1))2[A1]. (It follows that any M2q with (t; (Al))2 [Al] / 0 has order 8 in



the B Jy A-cobordism group II2 (v ,), in the notation of [7].)

Our results in this section will be based on the following: Let f: X —>

K(Z2, a) such that / (i) = x. Then I/: IX —> Ik(Z q) is the suspension of /

and 21/ is the sum of 1/ with itself using the suspension structure of IX.

(3.1) Theorem. The functional square Sq?¿, is defined and has indeterminacy

0, so that Sq«¿S Hq+1ilKiZ2, a); Z^ — H2q+lilX; Z¿, and Sq92^(I(t)) = I(x2).

Proof. Recall that if a, fe: IX —» V, (a + fe) = a + fe , so that with Z coef-

ficients, (2a) = 2(a ) = 0, and image (2a) = 0 in Z -cohomology. Further

Sq'*+ (It) = I Sq9 + (t) = 0 since dimension i = q and Sq9+ annihilates elements

of dimension < q + 1. Hence Sq2¿, is defined on Ii.

In IX, Sq«+1I = ISq^1 implies Sq«+I: HqilX; Z¿ - tf2<? + 1(IX; Zj) is zero, for the

same dimension reason. Since image (25./) = 0, as above, it follows that the in-

determinacy of the functional Sq9 + is zero (compare [7, (1.1)]).

To calculate the value of Sq2¿,(£(()) we consider the map 2/: X —» KiZ q)

defined as follows:

(3.2) X -^ XxX^L KiZ2, q) x KiZ2, q) JL« KiZ2, a)

where A(x) = (x, x), x e X is the diagonal, p (z) = t ® 1 + 1 ® i, so p defines

the H-space structure of K = KiZ a). Clearly (2/) (t) = 2x = 0 since we have

Z coefficients and hence 2/ is homotopic to a constant.

Suspending the sequence (3.2) and utilizing the homotopy equivalences

I(X x X) Si IX V IX V I(X A X) and I(K x K) ^ IK V IK V I(K A K) (cf.

[7, (1.5)1), we get a commutative diagram:

IX — M-, I(X x X) -^11-1 2(K x K) -ÎK

(3.3) 1, + 1, + ZA 1 + 1 + h(ß)

ix v ix v Kx a x) livx/va/A/^ lK v lK v 1(K A K)

where 1 .: IX — IX V IX V K.V A X) is the inclusion into the z'th tetm of thei _

wedge, i = 1, 2, A = sA, s: X x X —» X A X is the natural map collapsing X V X

to a point, and hip): K/C A K) —► IK is the Hopf construction on p (cf. [7,

(1.5) and after]).

Recall that for a suspension map co = la, 1/1 -^ Iß -*-* c we have that

(c/> + t/z)o) = rySw + Úicú (which is not true for an arbitrary map oj). Hence the com-

posite map in (3.3) along the lower level gives us 1(2/) = 1/ + 1/ +

fe(pt)(I(/ A /))(IA). Now 2/ 5? 0 implies 1(2/) =* 0 and thus we have shown

(3.4) 2i2f) St - bipMlif Af))ilA).


206 WILLIAM BROWDER

(3.5) Lemma. Let A -^ B —• C be maps. If Sq9+1(x) a72a" Sq9*1 are defined

with indeterminacy zero, then

SqV>) = «*Sq^Hx).

The proof is routine from the definitions.

Now we recall that by the argument of [26, (5.3)] we have

(3.6) Sq9 + 10(2(í)) = 2(tAí).

If (/>: 2X —» V, - </> is the map </> o r where r: 2X —» 2X is given by r(x, t) =

(x, -/), for 2X = X x 1/iX x 0 U X x l), so r*= (- l). Hence by (3.5), Sq9^ =

2-*Sq9^+1 = Sq9^1. Therefore we have from (3.4), (3.5) and (3.6)

Sa9+1 (2z) = Sa9+1 - (2t)Sq 2(Ï/)W" öqA(M)(2(/A/))(2A)

= (2Ä)*(2(/A/))*Sq9¿,(2t) = (2A)*(2(/(A/))*(2(z At))

= 2(Ä*(/*z A/*t)) = 2Ä*(x Ax) = 2(x2).

This completes the proof of Theorem (3.1).

As a corollary one gets the following result due to M. Barratt. Let X =

S" u e"+1 where a: S" -» S" is the map of degree 2. Then X , & 2X ando. re 77 +1 77

thus X is a suspension for 22 > 1. Hence we may add maps.

Corollary (M. Barratt). 2(/) is not nullhomotopic where I: X —» X z's the

identity, 22 > 1.

Proof. Take /: X —» KÍZ2, l), / z = x, the generator of H (X,; Z2). Since

x ^ 0, by (3.1) 22/ is detected by the functional Sq , and hence 22/ is not

nullhomotopic. By suspension, 2(2"/) is detected by Sq also and 2(2"/) is

also not nullhomotopic. But 2(2"/) = (2"/)(2(/)) and hence 2(/) is not null-

homotopic.

Let Al 9 be a manifold oriented in a cobordism theory with v , = 0, with

u/2i?+fe a un¡versai manifold for this theory, for dimension 2q. Let g: W/dW —>

2 A1+, c: Al —> W be the spectral orientation and classifying maps respectively,

so that h = (2 c)g: W/dW —» 2 W+ is the spectral cobordism invariant of M

(see §1).

(3.7) Lemma. Let x £ HqiM; Z.,). Then in 2A1 = AI U Al,

xpix ©x) = x2[Al],

where x ® x £ HqiM; Z.,) + H9(A1; Z2) = HqiM U AI; Z.,).



Proof. By (2.9), g2 = g + g- W/dW —• 2kM + V 2kM+ is the spectral orientation

map for 2M. Hence g* Ifc(x © x) = g*I*x + g*I*x = 0 so that x © x e (ker 77*)*,

(77 = g I ), and i/z(x © x) is defined.

Now if 8: M+ — K(Z2, a), ¿5*t = x, then S u8: (M U M)+ -*X(Zr q) repre-

sents x © x, (z5 U 5) í = x © x, and

2ki8 u 8) = Ife5 V 1*5: S*Af + V IfczM + -» 2*K(Z2> a).

Hence (I*(S U S))g? is given by

uz/(9Hz_=«, £*M+ V 2*A1 +_2^!l, 2*/<(Z2, a)

so that the composite is equal to 2((I <5)g). We may assume k is very large so we

are in the stable range, or we may suspend once to make g a suspension map, so

that 2ii2k8)g) = i2lk8)g. Then

Sq9+1 . (2fc() = Sq* + 1. dM = g*Sq'ï+I, (IM2((Zfe8)g) (2Zfe8)g (2Z*S)

= g*2kii8\)2) = g*I*(x2), by (3.5) and (3.1)-

Hence

^(x©x)=(Sqq+1, (SM)1>12(2* 8)g

= g*I*(x2)[W] . (iV)gJW] = (I*x2)(I*[M]) = x2[Ml,

(since gjH] = [M x Rk] = I*[/M]). This completes the ptoof of (3.7).

(3.8) Theorem. ki2M) is defined if and only if ic*y)2[M] = 0 for all y £

HqiW- Z2).

(Note that if kiM) and ki2M) are defined then *(2zVI) = 2kiM) = 0 by (2.14);

in fact kinM) = nkiM) by (2.13).)

Proof. By (2.9) if c2: M U M — W, then im c* = \ic*iy) © c*iy)), y£ HqiW; Zjl

By (2.6), ki2M) is defined if and only if i/>(im r2) = 0, since >72r2 = fe = (2fe) =

2fe* = 0 by (2.9) so that im c* C ker 77*. By (3.7), ipic*y © c*y) = (c*y)2[M], and

(3.8) follows.

Now we have the equivalent formulation:

(3.9) Corollary. ki2M) is defined if and only if, for M, v ÍM) e ker 77 , or in

other words, h v =0.1


208 WILLIAM BROWDER

Proof. By (3.8), ki2M) is defined if and only if (c*(y))2[Al] = iv u c*y)[Al] = 0

for all y £ HqiW; Z.,), where vq £ /79(A1; Z2) is the oth Wu class (see §1). In

other words M2AI) is defined if and only if all characteristic numbers involving v

ate zero. By (2.16), all characteristic numbers involving v are zero if and only

if h v =0. But h - r¡ c , so h v = 0 if and only if 27 (c v ) = 77 iv (Al)) = 0.q 1 q ' 1 q 1 q

This proves (3.9).

If M2A1) is defined we would like to calculate its value. If MAI) is defined it

follows from (2.14) that M2A1) = 2zé(Al), so we must consider the case where kiM)

is not defined.

(3.10) Theorem. Suppose k(M) is not defined, but that k(2M) is defined. Then

k(2M) = yj(v (Al)).

(Note that by (3.9), k(2M) is defined implies that v (Al) e ker 27 , so that

yj(v (Al)) is defined.)

(3.11) Corollary. k(4M2q) = (v (Al))2[Al].

(A similar result was obtained independently by E. H. Brown [25]-)

Proof of Corollary. The spectral cobordism invariant h. of 4A1 is given by

h4 = 4h, by (2.9), so that ker(2/>)* = H*(IkW + ; Z2) and Sq9*1 = 2Sq9+1 = 0, so

so that M4A1) is defined. If M2A1) is defined, then by (3-9) v (Al) e ker 27 r\

im c so that (v (M)) [Al] = 0, since im c annihilates ker 27 by (2.2). But by

(2.14), M4A1) = 2(M2A1)) = 0 if M2Al) is defined.

If M2AI) is not defined, then M4Al) = ipiv Í2M)) by (3.10) and v Í2M) =

v (Al) © v (Al) £ HqiM; Z.,) + H9(AI; Z2). Hence by (3.7), yjiv qi2M)) =

ifiiv (Al) ®v (Al)) = iv (A1))2[A1] and M4AI) = if/iv Í2M)) by (3.10), which proves

(3.11).Proof of (3.10). We suppose that MAl) is not defined but M2A1) is defined, so

that v (Al) e (ker 27*)9 by (3.9). Let R = (ker 27 n im c )q = radical of ( , ) on

ker 27 , and note that yj \ R is linear, 1/7 | R: R —» Z Let R = ker iyj \ R), so that

R' is of codimension 1 in R. Choose a complementary subspace N to R in

ker 27 , so that ( , ) is nonsingular on N, and choose a complementary subspace

A to ker 27 in HqiM; Z ) such that A is orthogonal to N (i.e. let A C annihila-

tor of N in W9(A1; Z2)). Then H9(A1; Z 2) = N + R + A, and since ( , ) is non-

singular on N, it is nonsingular on R + A and these two subspaces are orthog-

onal. Since A O ker 77 = 0, 27 | A is injective.

Then in 2A1, H"(2AI; Z 2) = r/9(Al; Z ¿ + W9(A1; Z.,) so H9(2A1; Z j = N + N +

R + R + A + A. Now 77*: W9(2A1; Z2) — W9+fe(U'/(9W; Z2) is given by

77* = 77* + 77*, by (2.9). Hence ker r]2 = ker 27 + ker 27 + AA where

A: rj9(AI; Z2) -> HqÍM; Z2) + HqiM; Z2) is given by Ax = x © x. Similarly



im c = A(im c ); and since q2c2 = fe-, = 2fe =0, im c C ker 772 and im c =

radical of ( , ) on ker 772, by (2.2). Hence ker 7/2/im c2 =N^ + N2 + R + A

where A = AA/im c* O AA, since R C im c*, identifying R + R/AR with R. It

follows that the two copies of N and (R + A) are orthogonal under ( , ) and ( , )

is nonsingular on each term. Hence if xp 2: ker 77 2 —» Z is the quadratic form,

c(<A2) = c(v>2|/V1) + c(./72|iV2) + c(v!z2|R + Ä). By (2.10), t/>2 | N. = xfi \ N, so that

ciip2 | N.) = ciyj \N), i = 1, 2, so that c(i/<2) = 2c(i/> | /V) + c(i/<2 | R + Ä) =

c(i/>2|R + Ä).

Now R = radical of yj on ker 77 is a self-annihilating subspace of

ker 77*/im c*, and if a © a, fe © fe e AA, then ia ® a, fe © fe) = (a, fe) + (a, fe) = 0

so AA and hence A are self-annihilating subspaces of ker 772 and ker 77 /im c2

respectively. Since ( , ) is nonsingular on R + A, it follows that R and A are

dually paired by ( , ), and hence any basis for R is half of a symplectic basis

for R + Ä. Since i/> | R = <A2 | R by (2.10), and yj\R' =0, it follows that if ä £ A~

such that, for r £ R', (r , a) = 0, then a basis for R together with x £ R such

that (x, a) = 1 are half of a sympletic basis for ker i72/im c and ciy/A =

i/> (x)t/>2(a) = z/r (a), since x 4 R' =ker(z/t|R).

Now ä 6 A is represented by a © a e AA and by (3.7), t/>2(a © a) = a2[M],

a £ HqiM; Z2). But a2[M] = (av9(M))[M], so that ifi2id) = (av<?(M))[Al]. Now ä

orthogonal to R is the same as a orthogonal to R , so that a generates the

annihilator of R' in A. Hence (ay (zM))[M] = (a, t> (zVl)) = 0 if and only if

v iM) £ R', or in other words ybiv (M)) = 0. Hence kilM) = ciyj2) = i/>2(a) =

(a, tz (M)) = ifjiv iM)), which proves (3.10).

Finally combining all the results of this section we get the following formula

which will be useful in §4.

(3.12) Corollary. Let M2q be such that (a) kiM) is defined, and (b)

iv (M))2[/M] 4= 0. Then for any integer n, kinM) is defined if and only if n é

2 (mod 4) and

kinM) = kiM) for n = + 1 (mod 8),

= kiM) +1 for n = + 3 (mod 8),

= 1 if n = 4 (mod 8),

= 0 if n = t\ (mo¿ si

The typical example of such M we use in §4 is projective space P q,

some M with a normal map into P 9.

)2[

(2.16)), it follows from (3.9) that ki2M) is not defined. For sM, h = sfe; so for

Proof. Since q ivqiM)) = rt*cv = b*v ¿0 and since iv ÍM))2[M] ¿ 0 (cf.


210 WILLIAM BROWDER

s odd, h = sh = h , and by (3.9) M2sAl) is not defined for s odd. Since h* =

2/3 = 0, it follows from (3.9) that kÍ4M) and M8A1) are defined, so by (3.11),

M4A1) = 1, and by (2.14) or (2.15), M8AI) = 0. Since i* = 4/3* = 0 and M4A1) is de-

fined it follows from (2.13) that M5AÜ is defined and M5Al) = MAI) + M4A1) =

MAI) + 1. Now ki- Al) is defined if MAI) is, and ki- Al) = MAI) (cf. (3.5)). Hence,

by (2.13), M3A1) is defined and M3Al) = ki- Al) + M4A1) = MAl) + 1. It follows from

(2.13) that M7A1) is defined and M7AI) = M8A1) + M- Al) = MAI), and periodicity

mod 8 follows also from (2.13).

4. Applications to surgery and fixed point free involutions. Let Al29 be a

smooth (or piecewise linear) manifold, and let ff , k » 2a, be a linear (or p.l.)

bundle over Al 9. A normal map (/, b) (see [8, §2]) consists of a map of degree 1,

/: Al' —> Al, Al' a smooth (or p.l.) 2i?-manifold, b: v —> ¿; a linear (or p.l.) bundle

map covering /, v = normal bundle of Al in S q + . We will use Al as the class-

ifying space, and the total space of Ei¿;~ + r~ ) = W, r = tangent bundle of Al,

as the universal manifold to define the spectral orientations and the Kervaire in-

variant. The existence of the normal map (/, b) ensures that v A£~) = 0 so that

W can be used to define the Kervaire invariant (see [8, Chapter 3l). In fact we

have the following:

(4.1) Theorem. With if, b) defining the orientation, MAI ) z's defined and if q

is odd and M is 1-connected or if M is nonorientable and 77.(Al) = Z then MAI )

is the obstruction to surgery, i.e., the obstruction to finding a normal cobordism

of if, b) to a homotopy equivalence.

This is proved in [8, §3] for q odd, Al oriented (no other hypothesis on Al).

If Al is nonorientable, then Wall [29] has shown the obstruction to surgery is the

Arf invariant of a quadratic form from H9(Al'; Z2) to Z2_ Using this, the same

proof as in [8, §3] holds.

Now we will use the results of §3 to construct normal maps with surgery

obstruction MAl') / 0 for particular kinds of manifolds Al.

Let Nm be a smooth (or p.l.) 222-manifold and T: Nm —> Nm a smooth (or p.l.)

fixed point free involution. Let Alm = N/T be the quotient manifold, £ the nor-

mal bundle of AT in Sm +k, and let (/, b) be a normal map /: Al' —> Al, b: v —> £,

as above.

We shall define the operation of multiplication by an odd integer on normal

maps as follows:

(4.2) Definition. Let 22 = 2s + 1, and define nif. b) = ig, c), g: M'in) —• Al,

A1'(t2) = 22AI' U - sN where 22AI' = Al' U-.-U Al', 22 times, - sN = (- /V)u---U (- N),

s times, - N is N with the opposite orientation. Then g = / on each copy of Al ,

g = 2-fold covering map p on each copy of N, so



degree g = tz degree / - s degree p = n — 2s = 1.

Then c = fe over each M, and a fixed map a" covering p over /V, where d is the

inverse of dp: r„ —> r„. Hence ig, c) is a normal map.

We will be interested in computing the invariants of tz(/, fe) in terms of the in-

variants of (/, fe). We will start with the spectral orientation map and spectral co-

bordism invariant.

Since £ is the normal bundle of M, it follows that W = M x R is the uni-

versal manifold. Then the map dp: tn —► r stably is the differential of an embed-

ding e: N x R —» M x R unique up to isotopy, using the Hirsch immersion the-

orem and general position. Then the embedding of M'in) x R C M x R is the dis-

joint union oí M' x Rk C M x Rk corresponding to (/, fe), and e: N x Rk C M x Rk,

translated by some values in R to make them disjoint. Hence it follows:

(4.3) Proposition. The spectral orientation map corresponding to nif, fe) is

the composite

VeVV2kM.^ V I*M + —-— Vl*M'+ V V2*/V +

7Z+S " S

where 8 is the map which pinches n + s — 1 copies of I M+ ro a point, g is

the spectral orientation of M given by if, fe), and r is the spectral orientation of

N given by d.

Now the classifying map of M'in) is / on each copy of M and p on each copy

of N so that the spectral cobordism invariant is the composite

l*M + -i V I'm^^V^vVI'n^^1«,TZ +S " 5

and hence we get

(4.4) Proposition. The spectral cobordism invariant of nif, fe), tz = 2s + 1, z's

given by

hiM'in)) = Tzfe(M') - shiN)

where hi ) denotes the spectral cobordism invariant of the corresponding manifold

and the suspension structure of I M . defines the sum.

Since 77 = 2s + 1 we can also rewrite Í4.4) as

(4.5) hiM'in)) = hiM) + s(2fe(zVi) - fe(zV))

which will be convenient later.


212 WILLIAM BROWDER

Now let suppose m = 2q and let us consider the following additional

assumption:

(4.6) If hN is the spectral cobordism invariant of N, assume

(a) b*N: Hqî2kM + ; Z2) — rz9+fe(2*A1 + ; Z?) is the zero map;

(b) kiN) is defined and M/V) = 0 (with the orientation induced by p and d).

(4.7) Lemma. Under the assumption (4.6), z'/ 22 is odd, kiM'in)) is defined

and kiM'in)) = M22AI').

This follows immediately from (2.13) and the fact that k(nM') is defined since

nM —> Al is of degree 1 mod 2 if 22 is odd.

(4.8) Theorem. Suppose Al, N, (f, b) etc. as above and suppose (4.6) is satis-

fied and (v (Af))2[Al] / 0. Then<?

kiM'in)) = MAI') for n = ± 1 (8),

= MAI') +1 for 22 = + 3 (8).

Proof. Since (/, b), f: Al' —» Al is a normal map, / is of degree 1, so v (Al') =

f*ivqiM)), /¿Al'] = [Al], and ivqiM'))2lM'] = if * iv qiM))2)lM'] = (vq(M))2f¿ M'] =

(z^(Al))2[Al] / 0. Then (4.8) follows from (4.7) and (3.12).

The obvious situation to apply these results is where Al = Pm, the real pro-

jective space, so N = Sm.

(4.9) Corollary. For any homotopy real projective space M q, there are nor-

mal maps (Midentity), 22 = + 3 (8)) of manifolds into Al q iwith its normal bundle)

of Kervaire invariant (i.e. the surgery obstruction) nonzero.

Proof. Simply take (/, b) = identity, and apply (4.8). It follows that for 22 =

+ 3 (mod 8), 2z(/, b) has Kervaire invariant 1.

(4.10) Corollary. There are smooth normal maps into real projective space

P2q with Kervaire invariant 1 (i.e. Tz(identity) for 22 = + 3 (8)).

This was known in the p.l. case by S. Lopez de Medrano [18] and Wall [30].

Let /: Mm —> Pm be a homotopy equivalence. Taking ç over P to be

/-1*(i/Al), we may define a normal map (/, b) into Pm, rf . Then if we make /

transverse regular on P2q C Pm, 2a < 222, we get a normal map (/ , b ): fq:

f-\p2l) -+P2q, b induced by b and /1 tubular neighborhood of /_1(P29). Let

us define

(4.11) Definition, k (Al) = kif~liP2q)) £ Z.,, i.e., kqÍM) = Kervaire invariant

of f~1iP2q) using the normal map (/ , b ) to define the orientation (as at the

beginning of §4).



(4.12) Proposition, k ÍM) depends only on the piecewise linear equivalence

class of M, in particular, it is independent of the choice of the homotopy equi-

valence f or the bundle map fe.

Proof. There is only one other homotopy equivalence f ': M —> P q up to

homotopy, i.e., the equivalence of opposite degree from /. But clearly / with any

choice of bundle map fe is normally cobordant to (/ u p, fe"), / U p: M U (- Sm)—»

Pm, p: Sm — Pm the twofold cover. Hence f'~liP2q) is normally cobordant

(using transversality) to / U p : /_1(P q) U (- S2q) —> P2q so by Lemma 4.7,

the Kervaire invariant defined by (/ , fe ) is the same as that defined by (/, fe"). It

remains to show it is independent of the choice of bundle map.

If fe, fe : v —» zf are two bundle maps covering the homotopy equivalence

/: Mm —» Pm, then there is a bundle map c: rf —» £ such that fe' = cfe. Therefore,

fe = c b , where subscript a denotes restriction to /" (P q) or to P q. It followsq q q' r L '

that the spectral orientation maps corresponding to fe and fe differ by the homo-

topy equivalence of I P q which is 5-dual to Tic): T(f) —> T(<f) so that the qua-

dratic forms and the value of kif~ (P q)) are unchanged.

Clearly if g: M. —> /VL is a p.l. equivalence, and if /, = g2g, then

g~ f2 (P q) = fT (P q) and it is clear that the Kervaire invariants are the same.

This completes the proof of (4.12).

(4.13) Proposition. // ttz = 4a + 1, then k2 iM) = desuspension invariant de-

fined in [lO].

This result has been proved by many people, e.g. Wall [30], Lopez de Medrano

[18], and Berstein [5].

(4.14) Proposition. k2 _ AM) = k iM), all a.

This has also been proved by many people including Lopez de Medrano [l8],

Wall [30], Orlik and Rourke and others.

One may ask the question, what sequences k2, k ,, • • • , k2 £ Z actually

arise as the Kervaire invariants k2-ÍM) for a homotopy projective space M. Lopez

de Medrano [l8] and Wall [30] have shown that for piecewise linear M, all such

sequences can occur. We shall study this question in the smooth case.

(4.15) Theorem. Suppose a sequence K = \k2, k ., • • • , k2 \ occurs as k2. =

k?.ÍMm), m = 4a + 1, Mm a smooth ior p.l.) homotopy real projective space. Then

the sequence K' = [K'2, • • • , k' \, k'2 . = k2- + 1, also occurs for some smooth M

ior p.l. M with the same stable normal bundle as M).

Proof. Let M" = Min), n odd, as in (4.2). Then (77/): Af(rz) -» Pm, and it

follows easily from the definition (4.2) that inf)~1iP2i) = (/" HP2i))in). Now it

follows from results of Wall [29l that in dimension m = 4q + 1 any normal map into


214 WILLIAM BROWDER

P is normally cobordant to a homotopy equivalence, so that Al" is normally co-

bordant to M , f : M —> Pm a homotopy equivalence. Using transversality, it

follows that f'~1(P2i) is normally cobordant to inf)~liP2i), so that k ÍM') =

kiif HP2i))(n)) = k2.(M) + 1, if 22 = + 3 (mod 8) by (4.8), which completes the

proof of (4.15).

Taking K = (0, • • •, 0) (with Al = Pm) we deduce from (4.15)

(4.16) Corollary. There are smooth homotopy projective (4q + 1) spaces Al

(normally cobordant to P49+1(t2) for n = + 3 (8)) such that k2.iM) = 1, all i, so

that in particular the desuspension invariant is nonzero.

Different constructions leading to a similar result have been found indepen-

dently by Berstein [5] and Giffen [13]. .Giffen's examples are on the Brieskorn

spheres, a result which we also found (see §6).

If we define /" (P1) C AI, f: M —> Pm, as a t dimensional characteristic sub-

manifold of Al, one may ask for what / a given M admits a characteristic sub-

manifold Q C Al which is a homotopy projective space. This is equivalent to an

analogous question about "characteristic" homotopy spheres in dimension t in the

2-fold covering space p: N —► Al. We remark that one may show, using the results

of Wall [30], Lopez de Medrano [l8], and Browder [9]:

(4.17) Proposition. The manifold Al of (4.16) admits homotopy projective

spaces as characteristic submanifolds only in dimensions t = 1 (mod 4).

We can use our technique to study the twofold cover p: N —> M.

(4.18) Proposition. For Al the manifold of (4.16), the twofold cover N repre-

sents a generator of bP 9+ , i.e. it is the Kervaire sphere which bounds a framed

manifold of Kervaire invariant 1.

Proof. By the construction, Ai is normally cobordant to Pmin), n = + 3

(mod 8). Now Pmin) C Pm+1in) and it follows easily that Mm C Wm+1, Wm+l

normally cobordant to Pm+1in). For if dU = M U Pm(r2), g: U -. Pm, then take

F(g (a)) = the total space of the line bundle g (a), a= canonical line bundle over

Pm, and take the union Vm+2 = F(g*(o)) U (Pm+1(r2) x /) with £((g|Pm(22))*(a))

identified with the normal bundle of Pm(n) C Pm+1(r2) x 0. Then dV =

W U Pm+1(72) x 1 and Mm C Wm+1 as required (compare the construction in [9])«

Now the normal bundle of Alm C U""+1 is (g | Al)*(a) which is the canonical

line bundle over Al so that the boundary is Nm the 2-fold cover of Alm. By (4.8),

Pm+I(22) has Kervaire invariant 1 if 22 = + 3 (mod 8), so that kiW) = 1. But /:

W —' Pm+1 with Al = f~ïiPm), /| Al a homotopy equivalence, so that it follows

that kiW) = kiW - Eia')) = 1 where a = normal bundle of MCW, and

diW - Eia')) = N (compare [8, (III.4.14)]). But f: W - Eia) - Pm +1 - Eia) = Dm + 1,



a disk, f\ N: Nm —► Sm a homotopy equivalence, so that it follows that W - E(a')

is a framed manifold of Kervaire invariant 1 whose boundary is N. This proves

(4.18).

In §5 we shall show that the N discussed above is a certain i4q + 1) dimen-

sional Brieskorn sphere so that (4.18) gives a new proof of the theorem due to

Levine and Milnor [20, §8] on the differential structure of these spheres.

The examples of (4.16) give smooth examples with homogeneous Kervaire in-

variants, i.e. with k2. - k2., all i, j < m/2. We conclude this section by con-

structing some inhomogeneous examples.

In his thesis, Sullivan [27j shows that there is a smooth A homotopy equi-

valent to CP with no characteristic 6-dimensional homotopy complex projective

space, and which has CP as a characteristic submanifold. An S -bundle ß

over A is then a homotopy P with P as a characteristic submanifold, so that

kABA = 0. However, Montgomery and Yang [2l] have shown that under these cir-

cumstances B has a characteristic homotopy P if and only if A has a charac-

teristic homotopy CP . Hence kAB ) = 1, since k is the obstruction to finding

such a characteristic homotopy P (see (4.13)). Hence K = JO, li can be realized.

Using the construction of (4.15), it follows that K = {l, 0J also occurs so that we

get:

(4.19) Proposition. For smooth ^-dimensional homotopy projective spaces all

combinations K = \k~, k A can occur as Kervaire invariants.

5. Smooth invariants of fixed point free involutions. In this section we apply

the results of §§1—4 to study free involutions on homotopy spheres. Recall that

if T: I7" —> Im is a fixed point free involution of a homotopy zzz-sphere I, then

the orbit space I/T is homotopy equivalent to real projective /«-space Pm.

(5.1) Proposition. Let M" have the homotopy type of P". Then there are

exactly two homotopy classes of homotopy equivalences of M" with P" deter-

mined by the degree (or the degree of the map of universal covering spaces

N" — Sn).

For a proof we refer to Olum [23].

(5.2) Lemma. Suppose M" is a smooth homotopy real projective m-space,

and let f: Mm —> Pm be a homotopy equivalence. Then ¡ivpm) = vMm, where

Vf., vp are the stable normal bundles in Smi^, k large.

Proof. From the result of Atiyah [2] and Hirsch [16], we know that / ivp)

is fibre homotopy equivalent to v„. If /: KOi )—►/() is the functor which sends

a stable linear bundle into its fibre homotopy type, Adams [l] has shown that

/: KOÍPm) —> ]ÍPm) is an isomorphism, so that / ivp) and 7/„ are linearly

equivalent, ptoving (5.2).


216 WILLIAM BROWDER

It follows from (5.2) that any smooth homotopy projective w-space A1m admits

normal maps into Pm, so that Alm can be oriented in the theory with universal

manifold Pm x R , k large, or Xm x R for any homotopy projective space X. We

shall study such Mm by means of this, through the spectral cobordism invariant

of (1.4) and the Kervaire invariants of §4.

Let Mm be a homotopy real projective m-space, Nm —> Alm its twofold cover-

ing, so that Nm is a homotopy //¡-sphere, and choose an otientation of Nm. We

will call this an oriented homotopy projective space. By (5.1) there is a unique

(up to homotopy) homotopy equivalence h: Mm — Pm suchthat h : Nm —> Sm is

of degree 1. By (5.2), h ivpm) = vM, so that it follows that h*ilrpm]) = ItMTn]

(where [ ] denotes the stable bundle). Hence, if we replace the composition Al —>

p77¡ ,- p77i x p ^ k » m, by an embedding, it follows that the normal bundle 27^ of

Mm CPm x Rk satisfies [27] = f*lrpm] - lrMm] = 0, and since k » 222, 27 is the

trivial bundle. Thus Alm x R is embedded in Pm x R as a tubular neighborhood

of the embedded Alm. Since k » m the homotopy class of h determines the em-

bedding of A1m up to isotopy in Pm x R , and by the tubular neighborhood theo-

rem, another embedding of Alm x R as a tubular neighborhood of Alm differs from

the given one by a linear map on each fibre, or in other words by twisting each

fibre by a map a\ Mm —> Oik) (normalizing to make it orthogonal). If we orient

Nm x Rk and Sm x Rk and require the induced embedding Nm x Rk C Sm x Rk to

be orientation preserving, then a: Mm —> SOik). Further changing a by a homo-

topy changes the embedding by an isotopy, so that up to isotopy our only ambi-

guity is iai e [A1m, SOik)] = KO_1(Al) (since k » 222).

(5.3) Proposition. KO" KPm) = Z-, for m / - 1 (mod 4), KO' l(Pm) = ZJ + Z

if m = — 1 (mod 4), where the Z is the image of KO~ (Sm) under the map j:

Pm — Sm of degree 1.

This is proved in [12].

In [26, Chapter IV] a map ß: Pm —» SO(m + l) is described with the property

that ß*o*w. = xl, where w. £ H'+ (BQ; Z 2) is the (2 + l)st Stiefel-Whitney

class, x £HliPm; Z.,) is the generator, o*: Hi + liBQ; Z.,) -^hKo^ZJ is the co-

homology suspension. Hence the Stiefel-Whitney classes w2, '"' wm+i of the

corresponding bundle over 2Pm are nonzero and therefore the bundle is not fibre

homotopically trivial. Hence we get

(5.4) Lemma. The class {ß\ £ K0-liPm) generates Z2 C KO-\Pm), J is a

monomorphism on the Z2 and the inclusion j: P' C Pm induces j : KO~ (Pm) —*

KO'Kp1) which is an isomorphism on Z2 and annihilates the Z factor.

Thus for a given homotopy projective space Al" and an orientation of its two-

fold cover N", if 22 é - 1 (4), there are exactly two choices of normal maps



M" —> X" (or X" x R orientations of M") where X" is a fixed homotopy pro-

jective space, corresponding to the two elements of K0~ (P"), while for tz = - 1

(4) there are other choices coming from the image of K0~ ÍS") in K0~ (P"). If

we take the spectral cobordism invariants in [I X" , I X"], then the set of ele-

ments corresponding to M" is an orbit under the action of the group [X", SOik)] =

KO-HX") on [IeX"+, 2kX"+], where SOik) operates on Sk in the right hand

I*X* = Sk A iX").

If cu 5" —» SOik), and /': X" — S" has degree 1 then the result of acting by

a/ on a map /: I*X^¡. — I^X^ is given by

(5.5) Lemma. // a represents x £ TTniSOik)), [/] £ [I*X"+, I*X"+], then x . [/] =

[/I + /***(/<*)) *»bere V nn+iSsk) ~* ^S"+k' lkx"J is lnducedby inclusion

Sk C 2kXn+, j* is induced by 2kj: I*X"+ — 2kS"+, and ] : niSOik)) — rrnîSk)

is the ]-homomorphism.

The proof is routine and we omit it.

It follows that if M", M" are smooth homotopy projective spaces, with N"

N" their (oriented) twofold covers, and fe ., fe2 £ [I X", I X"] their spectral co-

bordism invariants using X" a homotopy projective space, then M, is orientably

diffeomorphic to M, implies that fe, is in the orbit of fe2 by K0~ iX"). If

r: [2kX"+, I*X"+] — [I*X"+, IeX^] is the involution induced by the generator of

order 2 in K0~ (X"), then fe, = fe2 or rfe modulo ; iîjrr iSOik))). Thus we have

shown

(5.6) Theorem. The spectral cobordism irwariant defines a map from dif-

feomorphism classes of (oriented) smooth homotopy projective n-spaces to orbits

of [I*X"+, I*X"+] by KO-\Xn), so that different elements fe,, fe2 of

[I X", I X"] zrzay represent orientably diffeomorphic homotopy projective spaces

if and only if fe, = fe2 or rh2 modulo j iîjrr iSOik)). Further, if n / - 1 (4) then

this condition reduces to h , = fe2 or rh2.

(5.7) Lemma. [I*X"+, I*X"+1 = [I*X", I*X"+] + [5*, I*X"+].

This is a consequence of the general fact that the map z, + z': [X V Y, Z]—»

[X, Z] + [Y, Z] is an isomorphism, where i., z'- are the injections of X and Y

in X V V.

Now [Sk, I X"] = Z and the isomorphism is given by the degree of the

homology map. For the spectral cobordism invariants of homotopy equivalences

this degree is always 1, so we may concentrate our attention on the component in

[I*X", I*X"+].

(5.8) Lemma. For k very large ik > n + l), [I*X", I^X^ ^ [I*X", I*X"]

+ [I*X", Sk].


218 WILLIAM BROWDER

This follows from the general fact that the map 27,^+ 77 : [K, A x B] —>

lK, A] + lK, B] is an isomorphism. For the inclusion 2kX"+ = 2kX" V Sk C

2 X" x 5 induces an isomorphism for the functor [K, ] when dim K < 2k + 1

and, if k > n + 1, then dim 2kXn = k + n < 2k + 1.

In general we may study the group [2 X", 2 X^J by applying functors to it,

such as homology, or mapping spaces into the left variable 2 X", or mapping the

right variable 2 X" into another space. For example, take the map /: X" —> P°°

which induces the canonical line bundle on X, and suspend it, giving a map

llkX"+, 2kX"+] - [2*X"+, 2kP°°]. As 2*P°° is ¿-connected, /*: HkXn+, 2kP°°] -

[2 X", 2 P ] is an isomorphism, where / is induced by the suspension of the

inclusion /: X" — X"+. We denote by y the homomorphism y: [2*X"+, 2*X^] —

[2*X", 2feP°°].

(5.9) Lemma. Let p : N" —> Al" be the twofold covering of the smooth homo-

topy projective space and let hiN) £ [2 X", 2 X"] be the spectral cobordism in-

variant of N iusing the induced embedding coming from an embedding of M x R C

X" x Rk). Then rhiN) = hiN) and yihiN)) = 0 in [2feX", 2kP°°].

Proof. We give the proof for X = P", the proof in general being similar, r is

induced by iaf in the diagram:

(5.10)

/V"-L,/Vin

/

5"_JL,p"-1-,50(22 + 1)

,/c-l a.' , SOik) (see [26]).

Then the new embedding of N" x Rk C P" x Rk is induced by iafp = a'i'pf .

Since k » 22, 22 (P ) = 0 and thus i'p is homotopic to a constant, hence iafp

is homotopic to a constant and the new embedding is isotopic to the original.

Thus rhiN) = hiN).

Now hiN) is the composite map

2*P» £, 2*ai"+ - 2feN+ Li. 2feAl+ LÍ« 2kP"+

where c is the spectral orientation map for Al, s the Thom-Pontrjagin construc-

tion on N x Rk C M x Rk. Then yihiN)) is the composition with the map 2 /:

2feP^ — 2kP°°, where /: P"+ -> P¿. Then yihiN)) is represented by'



i2kl)iZkf)iZk'p)sc. But since N is 1-connected, Ifp: /V+ — P°° is homotopic to a

constant.

Hence 2kilfp) = (I*/)(!*/)(2*p ) is homotopic to a constant and thus yfe(/V)

= 0.

(5.11) Lemma. Let 1 = identity £ [2kXn+, I*X^]. Then y(l) = 2kln e[I*X", 2kP°°).

This is obvious.

Clearly t induces an involution (called again r) on [IeX", I*P°°] S

[2kXn, 2kP°°}.

Consider the diagram

[I*Pm, I*P°°] — [I*Pm, I*P°°]

(5.12)

[I*Pn, 2kP°°] -I— [I*P", I*P°°]

where z is induced by the suspension of the inclusion P" C Pm, m > n. It

follows from the commutativity of (5.10) that (5.12) commutes. Also z (I / ) =771

2kl wheie / : Pm —* P°° is the inclusion.n m

Now we recall [28] the following:

(5.13) Theorem (Toda). // k is large, n even, the order of the identity 1 £

[2kP", 2kP"] is equal to the order of the group IcOiP").

We recall also from [20]:

^ t \(5.14) Theorem. KOÍP") is a cyclic group of order 25( , generated by 1-x,

where 1 is the trivial line bundle, x is the canonical line bundle over P". Further

sin) = number of integers = 0, 1, 2, or 4 mod 8 which are < tz. (/tz particular

si4k + 1) = 2k + I.)

(5.15) Corollary. [I P", Y] is a Z .-module for any space Y, for n evert.

Proof. [I^P", Y] is a module over the ring [I*P", I*P"]. Since by (5.13)

[I^P", IfePn] is a Z s(n)-module, it follows that [I^P", Y] is also.

(5.16) Corollary. The order of ¡n £ [2kX", 2kP°°] is exactly 2s(n).

Proof. By (5.15), 2s(n)ln = 0. On the othet hand, (2*M*: KO-*(I*P°°) —

KO~fe(I*X"), /*(1) = 1, /*(x) = x (1, x being the trivial or canonical line bundles

over X" and P~), and (l\)* = !*(/*). Hence I*/* sends (x - l) £

KO-HlV00) SaHoiP00) into a generator of KO-*(I*Xn) SiKOiP"), by (5.14).


220 WILLIAM BROWDER

Now it is well known that (/V(2*'n))* = N(2*/n)*, so that (zV(2*/n))*(x - l) =

Nix - 1). Hence (¿V(2*M)*(x - l) ¿ 0 for N < 2s{n) by (5.14), so that N(lklJ is

not nullhomotopic for N < 2s("\ Hence order 2*/^ = 2s(n).

(5.17) Lemma. (r2*/ )* = + 2*/*: KO-*(2*P°°) - KCT H2*X").n — n

Proof. From (5.12) it follows that {i±rlkl ) = ri.1kl =r2.kl so that

(r2*/ )* = it rlkl )* fot m>n. Since r2 = 1, KO"*(2*P") = Z , . and77 * 772 ' 2S(">

K0'k(lkPm) — K0'*(2*P") is reduction mod 2*(n), and KO"*(2*Pm) =

KOiPm) is generated by / (l - x) (x the canonical line bundle over P°°) it follows

that the only involutions of K0~ (2 P") for all 22 compatible with this situation

are + 1, so that the lemma follows.

Using the construction in §4, for Mm a homotopy projective space, N its

(oriented) double covering, consider the normal map

f2r + x: ilr + l)AT u (- rNm) — AT,

given by the identity on each Alm, the twofold cover on each - Nm. If 722 = 4/ + 1

it follows from a theorem of Wall [29] that f 2 , is normally cobordant to a homo-

topy equivalence g2f ,: Mm(2r + l) —> Mm.

(5.18) Theorem. For 0 < r < s < 22l~ \ A14/+1(2r + l) is not normally co-

bordant to Al + (2s + l) iand a fortiori not orientably diffeomorphic).

Proof. By (4.5), the spectral cobordism invariant hÍMÍ2r + l)) = 1 +

r(2(l) - a) e [2 A1^, 2 Al™], where 1 = identity, a= spectral cobordism invariant

of the twofold cover Nm -> Alm. By (5.9), yia) = 0, so yhÍMÍ2r + D) =

(2r + 1)2*/ e [2*Alm, 2*P°°]. If M(2r + 1) and Al(2s + l) are orientably diffeo-m

morphic by (5.6), either hiMi2r + l)) = MAl(2s + l)) or rMAl(2s + l)). By (5.17),

rlll* = + lkl* , so thatm — m'

iybiMi2r + l)))* = (2r + 1)2*/* = ± (yh(M(2s + l)))* = + (2s + l)2*/* ,

Since /*(1 -x) generates KO(Pm) = RO_*(2*Alm) »Z„ „ it follows that this

7/1is impossible for0<r<s<2 ".

Note that by (5.16), 2*/* has order 22*+1 in [2*Alm, 2*P°°], so that one

cannot improve (5.18) without studying more delicate invariants, e.g. the part of

the spectral cobordism invariant in [2 Al ̂ S ].

One may use the same construction in other dimensions, but there will be a

surgery obstruction which is the Kervaire invariant (see (4.1)) if m = 2q, or the

Kervaire invariant of a codimension 1 submanifold if 277 = 4/ + 3 (see proof of

(7.1)). These obstructions are zero if and only if 2r + 1 = + 1 (8) (see (4.8)) so

we get, by a similar argument as above,



(5.19) Theorem. For m i 1 (4), (2/ + l)/VT U (- rNm) — Mm is normally co-

bordant to a homotopy equivalence g2^ ,: Mm(2r + l) —> Mm if and only if 2r + 1 = + 1

(mod 8), and different values of r give at least 2s(m)~i orientably nondiffeomorphic

homotopy projective spaces.

6. Involutions on certain Brieskorn spheres, smooth and PL classifications.

Let us consider the equation

(6.1) zd0 + z\ + z22 + ...+ z22q + l = 0

for z. € C, the complex numbers and d odd. Then (6.1) defines an affine alge-

braic variety with the origin as the unique singularity, of real dimension 4q + 2.

The intersection of the locus of (6.1) with the unit ball:

(6.2) ||z|| =E*.F. = 1

gives a smooth manifold 2,q+ of dimension 4q + 1. This manifold is a homotopy

sphere. The possibility of representing homotopy spheres in this way was dis-

covered by Brieskorn, and these representations are called Brieskorn spheres.

They have been studied by him and others including Hirzebruch and Milnor. A

beautiful account of this theory is given in [20].

The algebraic equations defining I ß + have natural symmetry which gives

naturally many groups acting smoothly on I ,9+ , which have been studied by

many authors (e.g. [17], [6]). Here we shall study the natural fixed point free in-

volution T: 24dq+1 — 24q+1 given by

(6.3) TU0'---'Z29+l) = (z0'-Zl'-22'---'-Z2z? + l)-

Clearly, if z = (z , ■ • ■ , z A is fixed under T, then z, = z, = .. . = z. , = 0

so that, from (6.1), zQ = 0 also, and z = 0. Hence T is without fixed points on

2-dq+ . These involutions were studied in the paper of Atiyah-Bott [4], and we

shall recapture and extend their results on smooth classification of them using

our methods. We shall also calculate the Kervaire invariants of them. Related

work has been done by Giffen ([13], [14]).

(6.4) Theorem. I^+1/7" is normally cobordant to P4k + lid). iSee §4 for the

definition of P4k + lid).)

Proof. If we modify (6.1) to

(6.1)' Piz) = zdQ + z2 + ...+ z22k+1 = e, e small, real,

and consider

(6.2)' ||z!|-£*.F.<l,


222 WILLIAM BROWDER

we get a manifold W + C C + , and it is easy to see that dW is equivariantly

diffeomorphic to 2^, with T defined by (6.3).

Now (6.3) defines an involution T: W —> W which has fixed points F =

i(A, 0, ••-, 0) £ W, where Xd = e}.

Consider N = Cx D4*+2 C C2*+2, N = {(zQ, ••-, z2fe + ,) such that

2. „z.z. < §1 where 8 « e, 8 teal. Then N is invariant under T and N =¡>0 i i _

N O W = d disjoint disks, a neighborhood of F. Then T \W — N is fixed point

free.

Since T is smooth, T\ N is smoothly equivalent to a linear involution, and

since it is free on dN it follows that T \ N is smoothly equivalent to the anti-

podal map on each disk. Hence (W - N )/T is a cobordism of 2 ,/T with

dP4k + l = dN'/T.

Now W - N' C C2* + 1 - C x Ö, where W - N' = P'l(t) O (D4*+4 - /V), Ö =

(0, ■ • • , 0) 6 C , P: C + —> C, í is a regular value. Since P is invariant under

T, PT = P, it follows that P induces P': C2* + 1 - C x Ô/T -» C and (W - A/')/T

= P'~lit) O V, where V = D4*+4- /V. Since T(zn, Zj, • • •, z2k j) = U0,-*,,..•,

-z2k+l), C2*+1-CxÖ/T= P4* + 1 x R3. Since (W - N')/T = P^K/) O V and /

is a regular value of P', (W - N')/T has a trivial normal bundle in P + x R ,

and hence it defines a "normal" cobordism between 2 ,/T and ^P + , except

that the inclusion maps of the boundary components into P + have degree d.

Take the 2-fold cover p: S + —> p + and make it regularly homotopic to an

embedding œ 54* +1 x / x Ri_ ' — P4* +I x R', / very large. This has degree 2

so if we take a connected sum along the boundary 2 ,/T of ÍW — N )/T with

(_ /)54*+1 x I x R''1 we get a normal cobordism of 2rf/T # (- /S4^+') = 2rf/T with

P4*+1(a/, which completes the proof of (6.4).

Applying (5.18) we obtain a different proof of the result of Atiyah-Bott [4]:

(6.5) Theorem. O22 the Brieskorn spheres 2 , + , with involution T: 2 , —»

2 ,, 2 ,/T , z's «oí normally cobordant to and hence not diffeomorphic to 'JL.jT,,

if 0<r < s <22k-\ d= 2r + 1, a"' = 2s + 1.

Giffen [14] has improved this result to 2 , for which we also have a proof,

which will Be given elsewhere.

Applying (4.15), (4.16), (4.17), noting that 2rf/T is normally cobordant to

P4*+1(a"), we get

(6.6) I hrorem.

*2.(24*^/T)=l ifjs±3 (8),

= 0 if dm + I (8),



atza" thus T admits homotopy l-spheres as characteristic submanifolds if and only

if d = + 1 (8) or I = 1 (4).

(This has also been proved by Giffen [13].)

From (4.18) we deduce a result of [20]:

(6.7) Theorem. The Brieskorn sphere 1^+1 bounds a manifold of Kervaire

invariant 1 if d = + 3 (8) and of Kervaire invariant 0 if d = + 1 (8). Hence Irf +

generates bPAk+2 for d s + 3 (8) and I^+l = 5«+l fof ¿ _ ± j (g)_

7. Embedding homotopy projective spaces. As an application of §§2, 3, and

4, and the results of [9] we give a theorem for finding embeddings of homotopy

real projective spaces. This theorem grew out of conversations with E. Rees and

is applied by him in [24] in the case where n = 7 and f = e .

(7.1) Theorem. Suppose <f z's a k-plane bundle over Pn, n > 5, k > 2. Sup-

pose p (rf) z's /z'fere homotopy trivial, where p: S" —» P" z's /fee 2-fold cover and

let a £ tt ATiÇ)) be such that ¿(a) O U = [P*] e H (p"; Z,), wfeere fe z's then +fe tz ' 2

Hurewicz homomorphism, U £ H (T(<f); Z2) /fee Tfeo/zz class, and [Pn] the funda-

mental class. Then there is a smooth M" C Sn+ + and a homotopy equivalence

f: Mn —» P" with the normal bundle of M equal to /*(£) + e ..

Proof. Since p f is fibre homotopy trivial, there is a ß £ tt kiTip (cf))) such

that fe(/3) D Upt ç = [S"]. Making /3 /-regular, we get a manifold g* C S" +* with

normal bundle z^0 and a normal map s: Q" —» Sn, e: Vq —> p ¿j. By using surgery,

we may assume Qn is 1-connected. If the Kervaire invariant of this normal map

(i.e. kiQ) with the orientation induced by e) is nonzero, we can get another

normal map s # s: Q # Q -> S" ti Sn = S" etc., and kiQ # Q) = 0. The composite with

p: S" —> P", and a map p ç —> cf gives us a map of degree 2, p': Q' —> P",

covered by e': i/Q, —» rf, where Q' C 5™^ with normal bundle vQI, and Mg') = 0.

Then ip', e ) represents a class ß' £ 77 ^(T(cf)), and since g' is 1-connected,

p'*=0 on HliPn;Z2) and hence p'*=0.

Now by making a: Sn+k — 7(f) /-regular on P" we get a map g: Nn —» P"

covered by a bundle map fe: f N — ¿f, where vN is the normal bundle of N in

S"4*. Now degree g = 1 (2) from the condition fe(cz) O U = [P"l in H (P"; Z,,),

so by taking the disjoint union with a number of copies'of the normal map p':

g —» P" we obtain a normal map (i.e. of degree 1) g': /V' —» P", fe': iy , —> cf,

represented by an element a £ tt ^(^df))* where a' = a+ j8'.

By the corollary of [9], if a' e n^îTiÇ)) suspends to the "normal invariant"

of some homotopy projective space M, then M C Sn+k+l with normal bundle

duced from f + f by the homotopy equivalence. But this is equivalent to th«

statement that the normal map into P" represented by a is normally cobordant

to a homotopy equivalence, i.e., that its surgery obstruction is zero, if tz > 5.

le în-

ic


224 WILLIAM BROWDER

From the results of Wall [29] there is no surgery obstruction if 22 = 1 (4) so in that

case we are finished. If 22 = 2a, the surgery obstruction is the Kervaire invariant

k(N') (see (4.1)). Hence if M/V') / 0, take the new normal map g : 3A/' U (- ö') —

P", etc. g = 3g' U p'•

Since p'*= 0, it follows from (2.13) that M3/V' U (- Q')) = ki3N') + kiQ1) =

k(3N') since Q' was constructed to have k(Q') = 0. Then by (4.8), M3zV') =

k(N') +1 = 0, so that the surgery obstruction is zero for the normal map g and the

proof is finished as before using [9l«

If 22 = - 1 (4), then we use the same process as above to construct ö. e

27 ATiç)) so that the normal map induced into Pn~ has surgery obstruction zero.

The remainder of the map is a normal map into a disk D" which is a homotopy

equivalence on the boundary, so it can be made normally cobordant to a homotopy

equivalence leaving the boundary fixed. Thus this normal map into P" is normally

cobordant to a homotopy equivalence (compare the proof of [9] and of (5.19)), and

again we are done.

BIBLIOGRAPHY

1. J. F. Adams, On the groups J(X). II, Topology 3 (1965), 137-171. MR 33 #6626.

2. -, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603-632.

MR 25 #2614.

3. M. F. Atiyah, Thorn complexes, Proc. London Math. Soc. (3) 11 (1961), 291-310.

MR 24 #A1727.

4. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes.

II. Applications, Ann. of Math. (2) 88 (1968), 451-491. MR 38 #731.

5. I. Berstein, Involutions with nonzero Arf invariant, Bull. Amer. Math. Soc. 74

(1968), 678-682. MR 38 #5225.

6. G. Bredon, Exotic actions on spheres, Proc. Conf. on Transformation Groups (New

Orleans, La., 1967), Springer, New York, 1967, pp. 47-76. MR 42 #1146.

7. W. Browder, The Kervaire invariant of framed manifolds and its generalization,

Ann. of Math. (2) 90 (1969), 157-186. MR 40 #4963.

8. -, Surgery on simply connected manifolds, Springer-Verlag, Berlin and

New York, 1972.

9. -, Embedding smooth manifolds, Proc. Internat. Congress Math. (Moscow,

1966), "Mir", Moscow, 1968. MR 38 #6611.

10. W. Browder and G. R. Livesay, Fixed point free involutions on homotopy spheres,

Bull. Amer. Math. Soc. 73 (1967), 242-245. MR 34 #6781.

11. E. H. Brown, Jri, The Arf invariant of a manifold, Conf. on Algebraic Topology

(Univ. of Illinois at Chicago Circle, Chicago, 111., 1968), Univ. of Illinois at Chicago

Circle, Chicago, 111., 1969, pp. 9-18. MR 40 #4964.

12. M. Fujii, Kçj-groups of projective spaces, Osaka J. Math. 4 (1967), 141 — 149.

MR 36 #2143.

13. C. H. Giffen, Desuspendability of free involutions on Brieskorn spheres, Bull.

Amer. Math. Soc. 75 (1969), 426-429. MR 39 #2168.



14. C. H. Giffen, Smooth homotopy projective spaces, Bull. Amer. Math. Soc. 75

(1969), 509-513. MR 39 #964.15. A. Haefliger and V. Poenaru, La classification des immersions c'ombinatoires,

Inst. Haute Études Sei. Publ. Math. No. 23 (1964), 75-91. MR 30 #2515.

16. M. W. Hirsch, On the fibre homotopy type of normal bundles, Michigan Math. J. 12

(1965), 225-229. MR 32 #1724.

17. F. Hirzebruch and K. H. Mayer, 0(n)-Mannigfaltigkeiten, exotische Sphären und

Singularitäten, Lecture Notes in Math., no. 57, Springer-Verlag, Berlin and New York,

1968. MR 37 #4825.

18. S. López de Medrano, Some results on involutions of homotopy spheres, Proc.

Conf. on Transformation Groups (New Orleans, La., 1967), Springer, New York, 1968, pp.

167-174. MR 40 #6564.

19. J. Milnor, Microbundles and differentiable structures, Princeton University,

Princeton, N. J., 1961 (mimeographed notes),

20. -, Singular points of complex hypersurfaces, Ann. of Math. Studies, no. 61,

Princeton Univ. Press, Princeton, N. J.; Univ, of Tokyo Press, Tokyo, 1968. MR 39 #969.

21. D. Montgomery and C. T. Yang, Free differentiable actions on homotopy spheres,

Proc. Conf. on Transformation Groups (New Orleans, La., 1967), Springer, New York,

1968, pp. 175-192. MR 39 #6354.

22. C. Morlet, Les voisinages tubulaires des varietés semi-linéaires, C. R. Acad.

Sei. Paris Se'r. A-B 262 (1966), A740-A743. MR 35 #4930a.

23. P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58

(1953), 458-480. MR 15, 338.

24. E. Rees, Embeddings of real projective spaces, (to appear).

25. C. P. Rourke and B. J. Sanderson, Block bundles. I, II, III, Ann. of Math. (2)

87 (1968), 1-28, 256-278, 431-483. MR 37 #2234a, b; MR 38 #729.

26. N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math.

Studies, no. 50, Princeton Univ. Press, Princeton, N. J., 1962.' MR 26 #3056.

27. D. Sullivan, Triangulating homotopy equivalences, Ph. D. Thesis, Princeton

University, Princeton, N. J., 1966.

28. H. Toda, Order of the identity class of a suspension space, Ann. of Math. (2)

78 (1963), 300-325. MR 27 #6271.

29. C. T. C. Wall, Surgery on compact manifolds, Academic Press, London, 1971.

30.-, Free piecewise linear involutions on spheres, Bull. Amer. Math. Soc.

74 (1968), 554-558. MR 36 #5955.

DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY

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